Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative polynomials with restricted zeros

Abstract

Sharp Remez-, Nikolskii-, and Markov-type inequalities are proved for functions of the form

$$f(z) = \left| \omega \right|\prod\limits_{j = 1}^m {\left| {z - z_j } \right|^{r_j } } \left( {\omega ,z_j \in C;0< r_j \in R;j = 1,2, \ldots m} \right)$$

under the assumptions

$$\sum\limits_{j = 1}^m {r_j \leqslant N} and\sum\limits_{\left\{ {j:\left| {z_j } \right|< 1} \right\}} {r_j \leqslant K,} 0 \leqslant K \leqslant N.$$

The Remez- and Nikolskii-type inequalities are new even for polynomials of degree at mostn having at mostk (0≤k≤n) zeros in the open unit disk.